We present an X-ray study of our nearest starburst galaxy IC 10, based on XMM–Newton and Chandra observations. A list of 73 XMM–Newton and 28 Chandra detections of point-like X-ray sources in the field is provided; a substantial fraction of them are likely stellar objects in the Milky Way due to the low Galactic latitude location of IC 10. The brightest source in the IC 10 field, X-1, has a mean 0.3–8.0 keV luminosity of ∼1.2 × 1038 erg s−1 and shows a large variation by a factor of up to ∼6 on time-scales of ∼104 s during the XMM–Newton observation. The X-ray spectra of the source indicate the presence of a multicolour blackbody accretion disc with an inner disc temperature Tin≈ 1.1 keV. These results are consistent with the interpretation of the source as a stellar mass black hole (BH), probably accreting from a Wolf–Rayet (W–R) star companion. We infer the mass of this BH to be ∼4 M⊙ if it is not spinning, or a factor of up to ∼6 higher if there is significant spinning. We also detect an apparent diffuse soft X-ray emission component of IC 10. An effective method is devised to remove the X-ray CCD-readout streaks of X-1 that strongly affect the study of the diffuse component in the XMM–Newton and Chandra observations. We find that the diffuse X-ray morphology is oriented along the optical body of the galaxy and is chiefly associated with starburst regions. The diffuse component can be characterized by an optically thin thermal plasma with a mean temperature of ∼4 × 106 K and a 0.5–2 keV luminosity of ∼8 × 1037 erg s−1, representing only a small fraction of the expected mechanical energy inputs from massive stars in the galaxy. There is evidence that the hot gas is driving outflows from the starburst regions; therefore, the bulk of the energy inputs may be released in a galactic wind.